Good evening everyone :
In the link here I found the following sentence : The Hodge conjecture predicts that the $\mathbb{Q}$ - linear span of the classes of algebraic subvarities in the cohomology of a smooth complex projective variety $X$ is given by the Hodge ring : $ \mathrm{Hdg} (X) = \displaystyle \bigoplus_p H^{2p} ( X, \mathbb{Q} ) \bigcap H^{p,p} ( X , \mathbb{C} ) $. The elements of this ring being called Hodge cycles. For elements of $ H^2 ( X, \mathbb{Q} ) $ this is a result of Lefschetz. It follows that if the Hodge ring of $X$ is generated by its elements of degree $2$, then the Hodge conjecture is true for $X$, and all algebraic classes represent intersections of divisors.
My question is :
What is concretly, the meaning of the last sentence as follows : if the Hodge ring of $X$ is generated by its elements of degree $2$, then the Hodge conjecture is true for $X$, and all algebraic classes represent intersections of divisors.
Thanks in advance for your help.
If The Hodge ring is generated in degree 1, then $Hdg(X)=<Hdg^1(X)>$, so any Hodge class can be written as a finite sum of products of (1,1)-classes, or equivalently, as Alex Youcis points out, products of classes of divisors. Hence the Hodge conjecture holds. The surjective map you are talking about is the map $Hdg^1(X)\otimes \cdots \otimes Hdg^1(X) \to Hdg^k(X)$, taking a sum of tensor products to its cycle class via intersection product.